The reform movement has promoted a shift in the central goal of
school mathematics: from the memorization of algorithms to conceptual
understanding. Number sense and function sense have been widely seen
as key components of this understanding. In this chapter, I propose
that:

Operation sense is the key link between number sense, function
sense, and the all-important symbol sense. Operation sense,
therefore, deserves to be at the heart of early mathematics.

A good way to support the shift towards understanding is with
a tool-based pedagogy: the use of manipulative, pictorial, and
electronic thinking tools. The use of such tools does not
guarantee understanding, but it does provide a good environment
for discussion, communication, and reflection, and can play an
important role in rethinking early mathematics, including the
necessary changes in teacher education.

The (enormously important) domain of arithmetic should not be
the whole of elementary school mathematics. An eclectic collection
of topics drawn from recreational math and elsewhere can provide a
rich source of contexts for worthwhile early explorations.

The implementation of this or any significant reform of
elementary school mathematics will require a political strategy.
The centerpiece of such a strategy should be the training and
hiring of math specialists throughout our elementary schools.

Making Sense

There is broad agreement in the math reform movement that schools'
emphasis on algorithmic mastery in both arithmetic and algebra is
misguided. The students trained under this regime fail to develop an
understanding of the underlying mathematics, and in fact soon lose
their grasp on the very skills that were intended to be the focus of
their education. To address this problem, the reform movement
proposes changes in direction and emphasis, in both curriculum and
pedagogy. These changes are often presented as a way to help students
develop number sense, and function sense.

The "sense" formulation is an improvement in our view of students
and learning. It pushes us to a view of the student as a thinker, a
person capable of understanding important mathematical domains, and
helps move our thinking away from the view of the student as a
programmable machine, a view that is consistent with the "skills"
emphasis, but inconsistent with the reality of young human beings.

In the domain of arithmetic, instead of spending hundreds of hours
trying to make children into a poor substitute for a 5 dollar
machine, we should shift to the development of number sense, mental
arithmetic, and an introduction to number theory. While much can be
learned by discussing and thinking about ways to carry out
calculations, accuracy and speed are no longer the goal:
understanding is. A similar shift is necessary in algebra. In "A New
Algebra" (Picciotto and Wah, 1993) I
articulated a proposal for new directions in secondary school
algebra. Not surprisingly, a necessary corollary to taking these new
directions is to make some changes at the elementary level.

A key tenet of the new algebra I propose is the emphasis on the development of symbol sense. A recent paper by Abraham Arcavi (1994) helps focus our attention on this. I fear symbol sense may be undervalued by some in the reform movement. In some cases, an overreaction to the failure of our current approach to Algebra may lead to a badmouthing of symbolic representation and manipulation. Such an over-reaction would only serve to perpetuate the current situation, but in a different way. We would go from algebra as gatekeeper through ineffective teaching of symbol sense (College Entrance Examination Board, 1990) to lack-of-algebra as gatekeeper through out-and-out non-teaching of symbol sense.

Many beginners in algebra confuse 2x, 2+x, and x2. This is not
merely a linguistic obstacle, but also a conceptual and mathematical
one. The notation would be easier to grasp if it were associated with
some meaning in the minds of the students. Another popular mistake
among high school students is to "distribute the square": (x+5)2 =
x2 + 25. It is easy to blame this on the weaknesses of Algebra 1,
but some questions remain: Do we think it is important for students
to understand this? If yes, how would one go about getting such ideas
across?

If students cannot perform simple algebraic manipulations
correctly, they cannot effectively pursue math, science, or
statistics. I am not talking about simplifying radicals within
fractions within radicals within fractions, but simple manipulations
like the one above, or like removing the parentheses in -(y-1). The
reality is that even simple manipulations are difficult to understand
for many, if not most students. That difficulty leads many educators
to despair about the possibility of teaching them, and makes it
tempting to conclude that these manipulations are not important. On
the other hand, the fact that these are seen as trivial by many of us
leads us to hope (foolishly) that they will take care of themselves
if we only engage the students in interesting math problems.

Students who cannot work with symbols are severely handicapped.
This is true whether their lack of facility with symbols stems from
being victims of the old-fashioned Algebra 1, or participants in a
reform curriculum that de-emphasizes symbolism to the point of
near-omission. Just like number sense has not and cannot be made
obsolete by calculators, symbol sense is a necessary part of math and
science, and cannot be made obsolete by technology. School algebra is
not the only conceivable symbolic framework, (computer programming
offers an interesting alternative), but it is a very important one.
In fact the computer programs beneath any new computer-based symbolic
representations themselves depend on the use of traditional algebraic
notation.

Operation Sense

Conceptually, number sense, function sense, and symbol sense
involve a substantial common component which I would like to call
operation sense. For example, take the sequence 5, 8, 11, 14, ...

Number sense should include the ability to recognize
repeated addition in this sequence, and its relationship to
multiplication.

Symbol sense should include the ability to express and
recognize the same thing in a form such as a+nd, or in this case
5+3n.

Function sense should include the ability to recognize
the relationship between that and the general linear function
y=mx+b -- here y=3x+5.

None of this is possible without a solid grasp of addition and
multiplication, their structural relationship, and their uses in
various applications. An understanding of operations is the
foundation of number sense, symbol sense, and function sense.

While it may be interesting to some to distinguish algebraic from
arithmetic thinking, it is useful to recognize operations as the
joint underlying foundation of both arithmetic and algebra (whether
you see algebra as being about functions, about structure, or like I
do see it as a combination of those). This has immediate curriculum
consequences, irrespective of the eventual resolution of the broader
theoretical questions.

Sample Lessons on Operations

When I taught elementary school, I taught many lessons to help my
students develop operation sense. Many of these lessons appear, in a
secondary school version, in Algebra: Themes, Tools, Concepts
(a secondary textbook I helped write -- Wah and Picciotto,
1994).

The Zero Monster:

The Zero Monster eats zeros, but there are no zeros in sight.
However, there are some cups and some caps [figure 1]. Cups and caps
cannot be turned over, but a cup can be combined with a cap to make a
zero, which the Monster can eat. For example, if there are seven cups
and three caps, you can make three zeros. After the Monster eats,
there are four cups left. So you can say that 7 cups + 3 caps = 4
cups.

This setup leads to various addition problems (all cups, all caps,
cups plus caps, caps plus cups, not to mention missing addend
problems of the various types. An interesting problem is "what Zero
Monster additions have answer 4 cups?" ) Zero Monster arithmetic is
of course isomorphic to the integers under addition.

This lesson is meant to complement, not replace other approaches
to integer arithmetic, such as the particle / antiparticle model, the
elevator model, the put in / take out model, the 1-D vector model,
the income / expense model, and the motions on the number lines
model. The question is not "which is the best model?" but "how do we
use and articulate multiple models when teaching important concepts?"

The McNuggets Problem

At McDonald's you can order 6, 9 or 20 Chicken McNuggets. This
allows you to order 15 McNuggets (6+9) but not 7 McNuggets. What
numbers can be ordered? What numbers cannot be ordered?

This problem, with a number theoretic flavor, is an example of what I believe is called the "stamp problem". (As in: What values can you get by combining any number of 23 cent and 32 cent stamps?) In the case of two stamps with relatively prime values, there is a nice result about the greatest value that cannot be obtained. However, at the elementary school level, that result is not as important as the experience with addition, multiplication, and their relationship that students gain while exploring it.

Another rich problem from the same domain that works well in
elementary school: In how many ways can each whole number (or
integer) be written as a sum of consecutive whole numbers (or
integers)? Again, the full answer need not be arrived at in order to
get a good mental workout and develop an understanding about numbers
and operations.

Mod Clocks

A mod clock [figure 2] is a talking function machine. If you say a
whole number, it responds with a whole number. For example, one mod
clock responded this way:

7 --> 2

11 --> 1

4 --> 4

17 --> 2

10 --> 0

Students try numbers as inputs, and the clock responds (in the
voice of the teacher) with a number from 0 to 4. The output is the
remainder when dividing the student's number by 5, but at first
students do not see this. They are able to predict the output based
on discovering a pattern relating the last digit of the input to the
output. However when asked what the special number of this clock is,
it is not unusual for a student to say "5". Matters are clarified
further by analyzing the behavior of other mod clocks, such as (in
rough order of difficulty) mod 2, mod 9, mod 3, mod 4, mod 6.

At some point in this process, it becomes useful to discuss
strategies for guessing the output. In some cases you can inspect the
number's last digit, in other cases the sum of the digits is useful.
In all cases, it is possible to count around the clock, and see where
one lands -- except of course for very large numbers, where such a
strategy is not likely to succeed.

Some mod clocks ding audibly before giving their outputs. For
example, in the case of the mod 5 clock:

7 --> ding, 2

11 --> ding, ding, 1

4 --> 4

17 --> ding, ding, ding, 2

10 --> ding, ding, 0

The number of dings corresponds to the number of times one passes
0 when counting to the input number, or in other words the quotient
of the division by 5. Again, students can be asked to predict the
number of dings, and worthwhile discussions can be had about how this
all works.

It would not be difficult to create a Mod Clock computer
microworld, but the activity works well without a computer.

This work can be followed up by defining addition and
multiplication mod 5 (or 9, etc). Students can be asked to look for
the identity element, the inverse of each element if it exists. They
can make addition and multiplication tables, solve missing addend
problems, and so on. Mod clocks make for explicit discussion of
operations, and for deeper understanding of the structures underlying
the real number system.

A related area to investigate is Calendar Math, where the current
month is declared to be infinite, and an arithmetic of the days of
the week is created. For example, assuming the month starts on a
Wednesday, the sum We+Th --> 1+2=3 --> Fr. Interestingly, you
would get We+Th=Fr even if you picked different dates for Wednesday
and Thursday, such as 8+9=17, or 1+9=10. The 3rd, the 10th and the
17th are all Fridays. (Of course, if the month did not start on
Wednesday, everything would be different.) An early question in the
investigation of Calendar Math is what day of the week would the
100th day of the month be? (The infinite month is needed in order to
make it possible to use late dates in additions or multiplications
without falling out of the month.) Again, one can define
multiplication, make tables, solve missing addend and factor
problems, find the identity and inverse elements, in short explore
the properties of these operations.

Magic Carpets

Magic carpets are the only means of transportation available to
travel among the lattice points. They come in various configurations
[Figure 3]. A carpet can be equipped with one to eight arrows: East,
West, North, South, North-East, North-West, South-East, and
South-West. For example, a carpet equipped with a North arrow, a
North-East arrow, and a South arrow makes it possible to go from the
origin to (3,2) in many ways. Two examples:

NE, NE, NE, S -- which could be abbreviated to (NE)3*S

S, NE, S, NE, S, NE, N, N -- which could be abbreviated to
(SNE)3*N2

Many questions can be investigated in this microworld, involving
notation, concepts, and the interaction of the two. For example: what
sequences of moves take you back where you started? what sequences of moves
are equivalent to each other? for a given destination, what is the
most economical way to get there? what are the rules for simplifying
sequences of moves? what are the "fastest" carpets? what are the
carpets that can get most places with the fewest arrows? what is the
greatest number of arrows a carpet may have and still not be able to
reach every lattice point?

The YZ game

The goal is to shorten strings of Ys and Zs, using the following
rules:

YZ = ZY

YYY can be erased

ZZ can be erased

For example: YZYZZYZZZYY

erase the first ZZ: YZYYZZZYY

erase the first ZZ: YZYYZYY

switch the first YZ: ZYYYZYY

erase YYY: ZZYY

erase ZZ: YY

YY cannot be simplified

After practicing by shortening YZ strings on the chalkboard,
students can be asked to predict the simplified version of a given
string. What are the possible resulting strings? How should we write
the empty string? What is the inverse of a given string? A "put
together table" for the six elements of the group can be filled in.
Exponential notation can be introduced, and strings like
Y9 Z9 can be discussed.

A follow-up is "the yz game", where the rules are:

z = yzy

yyy can be erased

zz can be erased

No commutativity this time. This is very un-intuitive, compared to
the previous example. Predictions of the simplified version of a
given string are extremely difficult to make, and strategies for
simplifying a string as simple as zyz are worth discussing.

Another follow-up: the "def" game (presented in story form in Wah
and Picciotto, 1994, p. 112). In this game, there are three symbols
rather than two, and the rules are:

dd=e

ee=f

ff=d

How many distinct elements are there? What is the identity
element? what is the inverse of each element? What are the powers of
each element? The fact that this group turns out to be isomorphic to
the (additive) calendar group discussed above is beyond elementary
school students' understanding, but may perhaps be discussed
profitably with their teachers.

A Conceptual Approach to Operations

The main purpose of the sample lessons presented above is to lead
students to pay explicit attention to operations and to their
structure. Most of them have a strong group theoretic flavor.
Surprisingly, (or perhaps not!), I have found that elementary school
students are far more open to these sorts of explorations than high
school students, who by then reflect the dominant culture and fear
that such work may be frivolous and lack "real world" applicability.
Of course, the lessons above only scratch the surface. Much more work
on operations needs to be done in elementary school, for example on:

undoing operations,

place value,

powers of ten,

order of operations,

rational numbers,

the effect of multiplying and dividing by numbers between zero
and one,

and so on.

By the time students reach secondary school, they should be
acquainted with the following ideas:

Models, interpretations, and uses of the four operations. (In
middle school include: squares and square roots; cubes; an
introduction to exponentiation.) In particular, this entails the
ability to recognize which operation can be used, and how, to
solve various problems, as well as the ability to recognize
whether one divides one number by the other, or vice-versa.

The structural relationships between operations, especially
the inverse relationship between addition and subtraction, and
multiplication and division, and also the distributive law. (In
middle school, include an introduction to the laws of exponents.)

Generalization of the concept of operation beyond arithmetic
by looking at concrete examples of groups. Familiarity with the
concept of identity and inverse element, commutativity and
non-commutativity. (This point is not intended to promote rote
parroting of formalistic rules. I hope the sample lessons showed
that these ideas can be approached at an intuitive and informal
level.)

In addition to lessons such as the ones mentioned above, the work
described in just about every chapter of this volume can help develop
these understandings, provided that the appropriate awareness is
present among the teachers. The purpose of the conceptual and
structural view of the operations is not so much to prepare students
for college-level abstract algebra, a goal that is only relevant to a
tiny fraction to our students. Rather, the purpose is to strengthen
our students' grasp of number, function and symbol, which in turn
will help prepare them for any sort of work in mathematics and
science, including especially algebra -- however you define it.

Speed and accuracy in executing traditional algorithms for
carrying out the operations should not be the goal of instruction.
Instead, discussion and analysis of student-generated and traditional
algorithms should have the goal to improve students' conceptual
understanding of the operations and of the number system. Such
discussion of division algorithms can for example reveal the
structural relationship between division and multiplication, and also
between division and repeated subtraction or addition.

Separating long division as a skill (not a very useful one these
days) from the concepts related to division is representative of the
approach I propose. The concepts will never be obsolete. A high
school parallel may help make the point: I do not teach my high
school students how to use a slide rule accurately and fast, since
that skill is obsolete. But I try to teach them why and how a slide
rule works, because this allows me to get to key conceptual points
about logarithms. The same approach applies to traditional arithmetic
algorithms. Being able to deconstruct them is worthwhile. Being able
to use them accurately and fast, less so.

The conceptual emphasis does not mean that we can ignore
"grammatical" issues, such as how to read expressions that involve
minus, or expressions that involve several operations. When working
on problems beyond a certain level of complexity, expressions such as
-(1 + (-2) - 3) will surface. This expression includes three minus
signs, each of which has a different interpretation. The last one,
between two sub-expressions, means "subtract," or "take away." The
first one, in front of an expression, means "the opposite of". The
middle one, in front of a positive number, means "negative". These
notational issues cannot and should not be avoided. Of course, to a
great extent, they are questions of convention and not of principle,
and should be treated as such, but that does not mean they are not
important.

Another example of a grammatical nature: students should be able
to tell that an expression like 2(3) + 1 is an addition, not a
multiplication, because the + is about the whole expression, while
the implied multiplication sign only affects the 2 and the 3. On the
other hand, the expression 2(3+1) is a multiplication, because the
implied multiplication sign is about the whole expression, while the
+ only involves the 3 and the 1. Again, this is a question of
convention, but it is related to an important structural concept: the
distributive law.

The Distributive Law

The most crucially important and difficult structural
understanding about operations at the elementary level is the
distributive law. Understanding it is fundamental to doing mental
arithmetic, as in: 24 * 30 = 600 + 120. This sort of mental agility
with numbers remains important in an age when electronic computation
increasingly replaces paper-pencil digit-pushing. (Of course there
are other important understandings that go into number sense, such as
especially place value.)

The distributive law is also a key concept in the development of
algebraic symbol sense. Abraham Arcavi (1994)
attempts to define symbol sense by discussing a number of examples,
several of which are built around the distributive law. For example,
understanding the distributive law allows one to see that the
equation (2x+3)/(4x+6)=2 does not have any solution. Or take the
problem: "Take an odd number, square it, and then subtract 1. What
can be said about the resulting number?" Writing the expression
(2n-1)2-1, and using the distributive law repeatedly, we get:

(2n-1)2-1 = 4n2-4n = 4n(n-1) = 8*n(n-1)/2

This allows the insight that the resulting numbers are triangular
numbers multiplied by 8. Arcavi's discussion of this example does not
focus on the distributive law, but on the ability of an expert to
read into the symbols: the second expression tells us that the number
is a multiple of 4, the third that since n and n-1 are consecutive,
it must be an even multiple of 4, ie a multiple of 8, and finally the
last expression that it is a triangular number times 8. However the
example also shows that it is difficult to separate interpretive
ability from manipulative facility. The insights the symbols give us
can only be there for us if we understand the underlying structure.

There has been a fair amount of bashing of factoring in the math
reform movement. This probably stems from the overemphasis that many
teachers and textbooks have put on the skill of factoring trinomials
at the Algebra 1 level. However, if students do not understand the
concept of factoring, they do not fully understand multiplication or
the distributive law, and they are essentially illiterate in the
symbolic realm. Once again, I suggest we make a distinction between
skill and concept. Factoring, as a skill, should be de-emphasized.
Factoring, as a concept, should be given more emphasis. It is a
particular case of reversibility, "an ability to restructure the
direction of a mental process from a direct to a reverse train of
thought" (Krutetskii, quoted by Rachlin, whose perceptive paper
"Algebra from x to why" (1987) should be
required reading for all who profess an interest in the learning of
algebra).

Manipulatives

"Geometry is the user interface for mathematics." (Attributed
to Bill Thurston.)

It is widely believed that working with numbers automatically
generates an understanding of such concepts as the distributive law.
Many teachers and textbooks describe algebra as the "natural"
extension and generalization of arithmetic. And of course, there is
some validity to that point of view. However the fact of the matter
is that many students have tremendous difficulty making the
transition to these ideas with variables, perhaps because they have a
shaky sense of number. In any case, work with well-designed
manipulatives can help build the necessary foundation to facilitate
the leaps to abstraction that are embedded and embodied in the
notation of algebra. For some students manipulatives provide an
important tool, for others, they provide a geometric context where
they can broaden and deepen their understanding, which is often only
mechanical mastery of algorithms.

Over the years Zoltan Dienes, Mary Laycock, and Peter Rasmussen
each contributed some useful ideas to the design of effective algebra
manipulatives. I took their work further and developed the Lab Gear,
a manipulative that has helped many teachers bring symbol-string
algebra to student populations who were previously frozen out of
secondary school math.

What some have called empirical algebra (work with Cartesian and
tabular representations in so-called realistic contexts) provides one
essential arena to help build students operations sense, but it does
not go far enough. Students' ability to understand operations is much
enhanced if the discussion of so-called "real world" problems is
complemented by the intelligent use of arithmetic manipulatives such
as Cuisenaire and base ten blocks, and algebraic manipulatives, such
as the Lab Gear. This is because those manipulatives give students
objects to think with and talk about. These objects provide an
additional language which helps students' number and symbol sense by
developing and building on their visual, tactile, and geometric
insights with a special focus on the operations. Used properly, they
help build a theoretical framework for algebra.

In the following section, I will introduce the Lab Gear, and
briefly demonstrate some key Lab Gear lessons, lessons that over time
and with discussion can lead to a solid understanding of the
distributive law. This approach gives much more weight to geometry as
a way to learn algebra than is currently fashionable in certain math
ed reform circles (for example among those who believe that all of
algebra should be seen through the lens of functions.)
Top

Base Ten Blocks and the Lab Gear

Base ten blocks are a model of the decimal number system. In the
model, place value is replaced by "shape and size" value [Figure 4].
Blocks represent ones, tens, hundreds, and thousands. The fact that
ten ones represent the same quantity as one ten, for example, is
clear from the very design of the blocks. However, seeing this is far
from automatic for young children! The blocks merely provide an
environment for mathematical activity and discussion, which can lead
to that understanding.

* Make a Rectangle

For example, making a rectangle with a certain number of units
leads to the interesting discussion of how many such rectangles are
possible. In the case where only one is possible, the number is
prime.

Making a rectangle with three tens and six ones can only be done
in the 3*12=36 arrangement. Doing a series of problems of this type
can provide good preparation for a discussion of how to multiply
one-digit numbers by two-digit numbers, and more generally for the
distributive law.

Making a rectangle with three x's and six ones is in some ways the
same problem, except that the multiplication is 3(x+2)=3x+6, a fairly
explicit example of the distributive law. Again, doing a series of
problems of this type can lay the groundwork for interesting
discussions.

"Make a Rectangle" is a straightforward question that all students
can understand. Actually finding the rectangle can be challenging,
particularly in the case of a trinomial. (Say the required blocks are
one x2, three x's, and two 1's.) Such a visual puzzle appeals to
many students. Being able to "read" the multiplication in the
resulting rectangle is helped by the use of the Lab Gear's Corner
Piece, which helps students see the dimensions of the rectangle. The
figure shows 3(2x+1)=6x+3 [Figure 6]

Some understandings about operations are built into such a figure:
that lining up two x's and 1 is a representation of x+x+1 or 2x+1,
that the area of a rectangle is a representation of multiplication of
its dimensions. These understandings cannot be taken for granted, and
their discussion is part of the essential purpose of the activity,
whose goal is only partially to factor the original expression.
Developing visual models for addition and multiplication is
pedagogically and mathematically very powerful. One cannot credibly
argue that students are better off not understanding these models
than understanding them.

On the other hand, I do not claim that merely doing these
activities will accomplish miracles. To develop an understanding of
the distributive law, and an ability to use it, takes a lot more.
"Make a Rectangle" must be followed up with a range of activities,
some using manipulatives, some not, and some making connections
between the manipulatives and the other representations. "Make a
Rectangle" starts with the area and asks for the dimensions. The
activity can be reversed: given the sides, what is the area? This
gives students a method for finding the product of 12 and 15, or of
(x+2) and (y+x+1). After building the multiplications with blocks,
students can sketch the block multiplication on paper, without
blocks, and then they can be shown the "multiplication table" format
for multiplying [Figure 7].

5x

+6

3x

15x2

+18x

+4

+20x

+24

Figure 7

These activities can be followed up or accompanied with work on
division (given the area and one side of a rectangle, what is the
other side?), with activities based on the volume of a rectangular
prism ("Make a box, write an equation about its volume"), as well as
with a study of patterns in squaring and cubing. Even after doing all
this work, there is no guarantee that students will have mastered all
the ideas that are involved. But the chances are much better than if
the geometric representation was denied them. As a follow-up on this
sort of work, in secondary school, algebra manipulatives can make
such previously arcane topics as completing the square considerably
more accessible.

* Perimeter

Another activity altogether with the Lab Gear, which is also very
relevant to the issue of understanding operations is the task: "find
the perimeter of this Lab Gear figure" [Figure 8].

This is a very useful activity for two reasons. On the one hand,
students find one part of it quite difficult, which reveals their
weak grasp of operations. (Specifically, the side of length x-1 often
eludes them.) On the other hand, this is a rich arena for alternate
strategies, and alternate solutions, which are interesting to
discuss. For example, is the perimeter 4x+2 or 3+3x+(x-1)? How could
both of these be right?

Some teachers object to this activity by bringing up that x and y
cannot be variable in this context, since for example, negative
values are meaningless. Indeed, this is a situation where x and y
each stand for a given (unknown) value. This is a common way to use
variables, and this activity provides a context where this subtlety
can be discussed with students.

Manipulatives Polemics

I cannot guarantee that the use of algebra manipulatives at the
elementary level will be helpful. I have used other manipulatives at
that level, and I have used algebra manipulatives at the secondary
level, but since I developed my algebra manipulatives while working
in a high school, I have no idea how they would be received by
younger children. All I can say is that I think this would be an
interesting and important area of research. My hunch, based on
reactions of upper elementary school teachers, is that algebra
manipulatives can be used profitably with students as young as 9
years of age. In any case, I would like to answer some of the general
objections to manipulatives that I have heard from various math
educators.

* Dimensions

Some educators object to this use of manipulatives because it
presumes an understanding of area (and volume) that not all students
have. That is a legitimate concern, but all it means is that we need
to learn how to help students develop those understandings. There are
various ways to deal with that (see for example some ideas in the
Smith and Thompson chapter in this
volume). Not only do manipulatives in no way inhibit the learning
of the concept of area -- in fact they provide both motivation and an
additional arena for the teacher to pay attention to this crucial
concept which has pay-offs and spin-offs all the way across the
curriculum. If the fact that some students have a weak understanding
of area meant we should avoid using area in teaching algebra, then we
couldn't use numbers in teaching anything, since some students have a
weak sense of number!

Beyond area and volume, some educators are concerned about the fact that in a Lab Gear problem like 3(2x+1)=6x+3, the x on the left side refers to the length of the block, while the x on the right side refers to its area [Figure 6]. Is this confusing to the students, they ask? Students do not usually know enough to be confused by this, but it is interesting for teachers to think about it.

The Lab Gear can serve as a model of dimensions with the following
convention: a thickness of one unit does not contribute to the
dimensionality of a block figure. From that point of view, x3 is a
model of a three-dimensional object, xy is a model of a
two-dimensional object, x is a model of a one-dimensional object, and
1 is a model of a zero-dimensional object. This is of course not
literally true: all physical objects, including Lab Gear blocks, are
three-dimensional. However this convention allows us to have a
consistent model. The x on the left is in an (x+2) by 1 by 1
arrangement, so we think of it as one-dimensional, and hence x refers
to its length. The x on the right is part of an (x+2) by 3 by 1
arrangement, so it is part of a two-dimensional figure, and x refers
to area.

In this model, the degree of a polynomial is the lowest dimension
that the blocks representing the polynomial can be arranged in. For
example, three x-blocks can be arranged as a two-dimensional
rectangle, or a one-dimensional line segment [Figure 9]. Since the
lowest dimension is one, 3x is of degree 1.

All this may be too subtle for elementary school students, but it
does not prevent them from using the manipulatives at an appropriate
level. After all, we don't expect children to understand how
calculators work, but that does not prevent the calculator from
serving a purpose in their education! The learning of any new domain
worth learning must necessarily rely on partial and incomplete
understandings at some times.

* Manipulatives as calculators

Yet others object to the use of manipulatives because they feel
uncomfortable at the fact that manipulatives can make previously
difficult work appear easy, and therefore can mask a lack of
understanding. That is a crucial insight. Deborah Ball
(1992) gives the example of students carrying out
a subtraction correctly with manipulatives, by following rules they
have memorized. However, once removed from the manipulatives, they
revert to their previous mistakes. The very same thing happens with
the example given above: students who can multiply binomials
correctly with the help of the Lab Gear, will later write
(x+5)2=x2+25. Does this mean that manipulatives should not be used? Absolutely not.

First of all, manipulatives obviously do not cause the mistake,
which happened before manipulatives even existed. Second, if a
student can do this multiplication correctly with manipulatives, upon
making this mistake when working with the symbols, mere mention of
the manipulatives by the teacher is often enough for the student to
think about it, perhaps draw a sketch, and make the correction
without any further guidance. In other words manipulatives can
provide a student-controlled home base to which they can retreat as
necessary, until they no longer need it. Obviously, the goal is to
get rid of the training wheels, to use Ball's metaphor, or the
scaffolding, in Mason's words, but that is not necessarily quick or
easy. Mathematical understanding can only be acquired through arduous
struggle. There is no royal road.

Moreover, the use of a tool to "get the answer" is not wrong in
principle. There are times when it is appropriate for students to use
technology (whether a calculator, a symbol manipulator, or
manipulatives) to get the answer to a subtraction, or to square a
binomial. The focus of a particular lesson, perhaps a "real world"
application that involves subtraction, may not be on the mechanics of
subtracting correctly, or on place value. In such a situation,
students need to get the right answer quickly, whether or not their
conceptual understanding of how it is computed is solid.

If we were to deny students access to tools that make this
possible, we would be joining forces with those who are trying to
prevent students from using calculators, and would have little chance
of getting very far. By the way, another way to think about the
"problem" that manipulatives make difficult things easy is that they
help weaker students pass required classes and exams -- a short-term
but real issue for them. They do not have the luxury we have to say
that those classes and exams --which most of us passed comfortably
and thereby gained access to the academy-- are garbage. As one of my
students put it: "The Lab Gear saved my butt!"

Of course, there should be times when the focus of the lesson is
on place value and on comparing strategies for subtracting two-digit
numbers. At such times, we may use manipulatives, or not, but in
either case, we need to devise lessons which will get to the
underlying concepts, not merely on "getting the answer." This is best
achieved through discussion.

* Conversations

Some educators object to the use of manipulatives because they do
not reflect mathematical abstractions accurately enough. For example,
it is pretty much impossible to represent certain fractions with base
ten blocks, and the Lab Gear blocks that represent variables are of a
fixed size, which seems to be a contradiction in terms.

However it is not the primary role of a transitional environment
for learning to be accurate. At the limit, the most accurate
representation we have of the concepts of algebra is the traditional
notation. If accuracy of model is all that is needed for pedagogical
effectiveness, our job would be easy! In fact, the primary purpose of
manipulatives is not to be accurate, but to offer an opportunity for
discussion among students, and for discussion between the different
cultures of student and teacher. Jeremy Roschelle (1996) conducts an
outstanding discussion of this issue in the context of designing
educational software for teaching physics. He correctly concludes:
"... rather than merely representing mental models accurately,
designers must focus on supporting communicative practices." In fact
it turns out that the limitations of the manipulative model provides
the spark for some of the best discussions in the classroom.

Here is an anecdote related by Pat Thompson:

A student suggested that the "thousand" base-10 block
could be replaced by an apple. The teacher asked what would then be
used to represent 100, 10, and 1. The student replied that apples
could be used for everything.

This is a great anecdote, which shows the power of manipulatives
as a conversation starter. The student may have made the suggestion
because he was grappling with the question "how could this one block
represent 1000?" -- a profound question. Or perhaps the student has
no clue about how 1000 differs from 100, or 10 or 1. In any case, the
student would not have been better off if the base ten blocks had
been excluded from the classroom. It is precisely because of their
presence that an opportunity for a discussion of how and why one
block can represent 1000 (or how and why 1000 is different from 100,
etc) arose. In most classrooms students do not have the opportunity
or the context to verbalize their thoughts and misunderstandings. A
single discussion is not sufficient, of course, but a fruitful
discussion of student-designed representations for our numeration
system could follow the student's suggestion about apples, using
apples, buttons or whatever. After such a discussion over a period of
days, the teacher (with the help of those students who already
understand it) may be able to re-introduce the base 10 blocks in a
way that would be more meaningful to the student. Clearly, while
understanding the model cannot be equated with understanding the
concept, not understanding base ten blocks is diagnostic of not
understanding numeration in base ten.

To take another example, consider the story of Adam, the fourth grader who found that "if you take two consecutive numbers, you add the lower number and its square to the higher number, and you get the higher number's square" (Bastable and Schifter, in this volume). As Bastable points out, writing this in algebraic notation is enough to prove the correctness of Adam's assertion. However, in fourth grade this route is closed, since to a nine-year old this amounts to meaningless gibberish. A priori, the only way at that level to check the validity of that assertion is to try it with various numbers, which is indeed what was done. But another way is open if we keep in mind the strategy used by the students in Rigoletti's third grade, who represented square numbers with square arrangements of unit squares. [figure 10] An extra level in the process of generalization could be seen in the Lab Gear representation of the same phenomenon.Such an arrangement could be arrived at through questions such as the following [figure 11]:

How would you represent two consecutive numbers with the Lab
Gear?

How would you represent the square of each of these numbers?

How would you represent Adam's conjecture?

A highly skilled teacher is able to generate worthwhile discussions with students with or without manipulatives. Manipulatives, because they are, in Papert's (1980) terminology "objects to think with" and objects to talk about, have the potential to improve discourse in more classes. They are often a teacher's first step away from traditional instruction, and they can lead to decisive changes in classroom culture: the introduction of collaborative work, the option for students to create exercises instead of merely solving them, an opportunity for the teachers to enjoy a new window on student understanding or lack of it, and so on.

* States vs actions

Jim Kaput (in press) argues that
computer-based "manipulatives" would have advantages over physical
ones, because of the computer's ability to record processes, or to
display variables with varying sizes. It may well be that such
software could be designed, tested, and distributed to schools along
with the necessary hardware. However, the reality is that physical
manipulatives will reach many, many more classrooms much, much
sooner. So it is useful to ponder whether what Kaput calls the
"eternal present" of manipulatives can be transcended with adequate
design, lessons, and teaching.

Kaput points out that when modeling 2+3=5 with unifix cubes, for
example, by the time one has combined 2 and 3 blocks together, one
sees 5 blocks, but the original 2 and 3 blocks, as well as the
process of combining them has vanished. True enough, though he admits
that color can help overcome this. For multiplication, the Lab Gear
corner piece allows students to see both the factors and the products
when multiplying, or factoring. This is a design solution. But more
significantly, the activity "add 2 to 3 using Unifix cubes" is just
about the least interesting activity imaginable in this domain. Using
Krutetskii's and Rachlin's insights, we can readily come up with the
following improvement: "What are all the ways that you can combine
sets of Unifix cubes in order to get a total of 5 cubes?" This
encourages students to keep records of which pairs worked, gives them
a chance to notice patterns (using one less red cube requires using
one more green cube, for example), and focuses their attention on the
process rather than (or in addition to) the result. It also provides
the springboard to a question for the top students in the class, who
can start investigating the question of how many partitions of 5 are
possible.

Kaput also argues that the computer makes it possible to have
"hot" links between representations on the screen. In other words,
when you manipulate symbols on the screen, images of blocks could
react to the manipulations as you go. This is nice, and some software
along these lines exists to show the connections between different
representations of functions. However, we should not be lulled into
thinking that an on-screen connection guarantees a connection in the
student's mind. The key question is not one of equipment -- it is how
to use the available equipment well. In spite of Papert's predictions
(1980), the technology of the last 15 years did
not bring about an educational revolution. I see no reason to believe
this will change in the next 15 years.

Tool-based Learning

Deborah Ball, in a brilliant paper (1992),
warns against the "magical hopes" that many teachers have about
manipulatives. She writes: "Manipulatives -- and the underlying
notion that understanding comes through the finger tips -- have
become part of educational dogma." She gives several examples of the
mere use of manipulatives failing to deliver understanding, and
concludes that manipulatives cannot be used effectively without
adequate teacher preparation, and without better understanding of how
children learn. She also acknowledges that "Manipulatives undoubtedly
have a role to play ... by enhancing the modes of learning and
communication available to our students." I strongly agree with
Ball's concerns and insights, and do not claim magical powers for
manipulatives. In fact, in my own teaching, manipulatives have always
been one tool among many. I use computers and calculators, graph
paper, and group discussion. Sketches on the chalkboard, whether
drawn by me or by my students, play a crucial role in my class, and
probably in any effective math teachers' class. The use of any one
tool is not a rejection of others!

On the other hand, teachers' interest in manipulatives provides us
with an opportunity to work with them. In the past few years, I have
been asked to introduce in-service and pre-service teachers to the
Lab Gear dozens of times. These workshops have given me the
opportunity to work with hundreds, perhaps thousands of teachers, and
the discussion inevitably goes beyond the technicalities of the Lab
Gear, to important mathematical, curricular, and pedagogical
questions, such as the shift in emphasis from "skills" to conceptual
understanding.

At any rate, over years of work with both students and teachers, I
have learned that:

Manipulatives are an extraordinary tool to help reach weaker
students, but that is not their only purpose: they are a useful
way to improve education in any math class.

Often, teachers are ineffective because of their own limited
understanding of the material. Manipulatives provide an
environment to teach math as well as pedagogy to teachers.

Manipulatives do not make math "easy", and teachers may need
to learn something in order to use them. The increased
understanding will serve them whether or not they use
manipulatives in their class in the immediate future.

There is no sense in using manipulatives in a "do as I say"
algorithmic model which only perpetuates antiquated pedagogy. It
is far more effective to use them as a setting for problem
solving, discussion, communication, and reflection.

Manipulatives should be a complement to, not a substitute for
other representations. In particular, Cartesian graphing and other
pictorial representations are extremely important.

Deliberate attention must be paid to help students transfer
what they know in the context of the manipulatives to other
representations, including symbolic, numerical, and graphical.
Transfer does not just happen spontaneously.

The purpose of manipulatives is to enhance discourse, and to
provide a locus for the more theoretical discussion of the numeration
system (in the case of base 10 blocks) or of the structures
underlying aspects of the field structure of the number system and
its algebra (in the case of both Base 10 blocks and the Lab Gear).
That theoretical dimension runs the risk of being buried or forgotten
if we go full-steam into the currently fashionable direction of "real
world" problems.

Fun Math

In most schools, early mathematics means arithmetic. Many
elementary school teachers are not comfortable with math topics
beyond basic arithmetic. Most people, including elementary school
students and their parents, are convinced that mathematics is
arithmetic. Until recently, the NCTM journal for elementary school
teachers was called The Arithmetic Teacher, revealing that
confusion even among the leaders in the profession.

Below, I list some "fun" domains for early mathematics. For each
domain, I hint at some interesting problems or materials, most of
which I have used with young children in my ten years as an
elementary school math specialist and classroom teacher. Obviously,
this is not a comprehensive curriculum, just an attempt at giving
some substance to the claim that a lot more than arithmetic is
possible in elementary school.

* Perimeter Patterns

Pattern blocks [Figure 12] are available in many elementary
schools. They can be used as a source of interesting perimeter
pattern problems.

For example, by making a figure with alternating squares and
hexagons, [Figure 13], we get the following sequence of perimeters:
4, 8, 10, 14, 16, 20, 22, ... The sequence is rich in patterns that
students can explore. What kinds of numbers can be found on this
list? What kinds of numbers cannot? What would the hundredth number
be?

A simpler sequence is obtained by adding the square and hexagon as
a unit, yielding the sequence: 8, 14, 20, 26, ... Simpler yet are
sequences built out of using only squares or only hexagons. Such
sequences are arithmetic, and students can learn to represent them
symbolically.

Another question from the same domain: Is it possible to create a
pattern block figure with an odd perimeter?

* Angles

The same blocks can be used to introduce the concept of angle,
which is notoriously difficult to introduce to tenth graders if they
haven't worked with it before. Here is a sequence of activities using
the pattern blocks. (Note: pattern block hexagons are yellow, squares
are orange, and so on.) These will take several days to complete.

In how many ways can you surround a point with pattern blocks
using only yellow blocks? using only blue blocks? using two
colors? etc.

There are 360 degrees around a point. Use this information to
find the measurement of the angles in each pattern block.

What is the sum of the angles in each of the pattern blocks?
Using one or more pattern blocks, make a polygon for which the sum
of the angles is greater than that for the quadrilaterals, but
less than that for the hexagon. Create a 7-gon, an 8-gon, etc with
pattern blocks. In each case, find the sum of the angles. What is
the pattern?

Polygon walks: give instructions to a classmate to walk each
pattern block shape. (Example: take a step forward, turn right 90
degrees, etc.) Give instructions to walk a two-pattern block
polygon. This activity can lead to turtle geometry work on
computers.

* Geometric Puzzles:

Geometric puzzles such as tangrams and pentominoes offer
opportunities for lessons on similarity, congruence, symmetry, area,
and more. (Picciotto, 1986. See also EDC's
Elementary Science Study _Tangrams_ unit from the sixties.)

I will give two examples from the world of pentominoes. Pentominoes are the twelve distinct shapes that can be obtained by joining five squares edge-to-edge [figure 14].

What rectangles can be covered exactly by one or more
pentominoes from one set ? (Pentominoes cannot be overlapped or
reused.) To get this discussion started, students can try to cover
rectangles such as: 3 by 5, 2 by 5, 2.5 by 6, 2 by 10, 3 by 6, and
so on. Students discover that pentomino rectangles must satisfy
these conditions:

whole number sides

an area that is a multiple of 5, between 5 and 60,
inclusive (except for area 10, surprisingly)

If you double (or triple) the dimensions of a given pentomino,
can the resulting shape be covered with pentominoes? How many
pentominoes are needed to do it?

Many other interesting questions can be asked about pentominoes,
and about the generalized polyominoes. A related domain is
superTangrams, shapes obtained by joining four isosceles right
triangles edge-to-edge. In this case, problem 2 above becomes a lot
richer and more complex, because in addition to doubling and tripling
the dimensions, it becomes possible to get similar figures with
area equal to twice or eight times the original area.

* Lattice geometry:

How many different-sized squares can one find in an 11 by 11
geoboard? This leads to a major hunt which includes experiences
about slope (how to find perpendicular sides if the vertices are
all on lattice points), and area (the area of a "tilted" square
can be obtained without the Pythagorean Theorem by subtracting
right triangles from "horizontal" squares). (See Picciotto and Wah,
1993 for more on this.)

Pick's formula relates the area of a polygonal shape with all
vertices on lattice points to the number of lattice points on the
circumference, and to the number of interior lattice points. With
some guidance and much exploration, students can discover this
formula.

Many interesting explorations can also happen at a more elementary
level, and others (leading to strengthen ideas about angles) are
possible on the so-called circular geoboard, which features pegs
spaced at 15 degree intervals on a circle, as well as a central peg.

* Graph Theory:

The Koenigsberg bridge problem, Euler's formula on the plane or in
polyhedra. The two-color theorem (what maps can be colored in two
colors? how can one change a three- or four-color map to a two color
map by adding edges?)

The discovery of Euler's formula, like Pick's, makes for
particularly interesting lessons, because they each involve three
variables instead of the usual two -- which makes for more protracted
investigations.

And there are more arenas for the development of early
mathematical thinking: programmable computer environments such as
Logo, Boxer, spreadsheets, etc; logic with attribute blocks, the game
of Set, etc.; art with tiling, symmetry, Escher and Islamic designs;
data analysis and probability; etc. I hope it is clear that children
need not be fed only a steady diet of number crunching. Plenty of
interesting mathematical work can be done in a wide range of
mathematical domains such as the ones listed above.

While I in no way want to diminish the importance of work with
numbers, we must acknowledge that by de-emphasizing multi-digit
paper-pencil arithmetical algorithms at the upper elementary levels,
we are opening up hundreds of hours for other mathematical work.
Instead of filling that with an attempt to teach the traditional
Algebra 1 course to younger and younger students (an unfortunate
trend in many school districts), we should seize the opportunity to
broaden the curriculum. The point is not that students need to know
Euler's formula in sixth grade, but that they need to be engaged in
worthwhile mathematical thinking at an appropriate level.

Conclusion

I have tried to articulate a vision of elementary school
mathematics that would be considerably richer than the current
fixation on arithmetic. It is also richer than some reform proposals.
The following maps may help us rethink early mathematics:

Where we are:

Arithmetic

---> "Algebra 1"

---> Secondary Math

Where we should go (with an emphasis on operations throughout):

Arithmetic and number theory

Quantitative reasoning and the rational numbers

"Fun Math" as described above

Manipulatives and computer programming
(to support the above)

---> Secondary Math

That vision stems from some political assumptions, and it has
political consequences.

The assumptions: mathematics education is not just about preparing
students for "practical" matters and helping the economy. It is an
important part of human culture of sense-making, and should be
introduced as such to all students from a young age. In addition to
being useful, math is fun and beautiful. We should not lose sight of
this as we attempt to make the curriculum more relevant through
greater reliance on applications.

The consequences: we need math specialists in all elementary
schools. It is not likely that this vision or anything like it can be
implemented without having people with at least a BA in mathematics
as well as some pedagogical sophistication involved in the front
lines. This is more important than any particular curricular
decision, because such people in the schools would transform the
daily experience of math education with whatever curriculum.

One could argue that a massive effort aimed at training elementary
school teachers, through both pre-service and in-service work, is
needed. I don't disagree with this, but one math specialist can work
with approximately ten classrooms, and provide an on-going presence
that will have much more impact on teachers than a one-shot training
session. That ten-to-one ratio also means that teacher-training
budgets can have an impact that is one order of magnitude greater.

Finally, a national campaign to put math specialists in all
elementary schools is a program that teachers and parents in all
communities could support even if they are not ready to make major
curricular changes. Who knows, they may be able to talk politicians,
funding agencies, and corporations into supporting such an effort.
Whether or not we succeed in making curricular changes in the short
run, a national corps of math specialists could be the communication
line (in all directions) between the mathematics community, the math
ed community, and the elementary schools' students, parents, and
teachers.

Sidney L. Rachlin. "Algebra from x to why: A
Process Approach for Developing the Concepts and Generalizations of
Algebra." In Wendy Caughey (Ed.), From Now to the Future.
Melbourne, Australia: The Mathematical Association of Victoria, 1987,
213-217.